"Despite this law moving forward, there seems to be some deeply entrenched and rather strong opposition," Whiteman said. "It doesn't seem like there's a lot of good news for the Obama administration. They still have a lot of work to [do to] win people over."

The survey found, however, the percentage of Americans who lack health insurance was 11 percent, the lowest rate since Bankrate began asking that question last August. Last summer, 15 percent of respondents reported not having health insurance, Whiteman said.

A Bankrate survey last month found 45 percent of respondents said they believed Obamacare should be repealed; 44 percent said it should be maintained, CNBC notes.

"I think that the law has never really recovered from the bungled rollout of the health exchanges," Whiteman said, adding House Republicans’ dogged attacks on the law "has to be having an impact on people's opinions."

The poll also found those who earned more than $50,000 annually were most inclined to have negative feelings about Obamacare's effect on the country, while those pulling in $30,000 or less tended to be positive.